An exact penalty approach for optimization with nonnegative orthogonality constraints

• 成果类型: Article
• 署名作者: Jiang, Bo; Meng, Xiang; Wen, Zaiwen; Chen, Xiaojun
• 署名单位: Nanjing Normal University; Massachusetts Institute of Technology (MIT); Peking University; Peking University; Hong Kong Polytechnic University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01794-8
• 发表日期: 2023
• 页码: 855-897
• 关键词: nonlinear-programming problems; Matrix Factorization; mathematical programs; optimality conditions; Newton method; algorithms; nonconvex; minimization; CONVERGENCE; SPARSE

摘要

Optimization with nonnegative orthogonality constraints has wide applications in machine learning and data sciences. It is NP-hard due to some combinatorial properties of the constraints. We first propose an equivalent optimization formulation with nonnegative and multiple spherical constraints and an additional single nonlinear constraint. Various constraint qualifications, the first- and second-order optimality conditions of the equivalent formulation are discussed. By establishing a local error bound of the feasible set, we design a class of (smooth) exact penalty models via keeping the nonnegative and multiple spherical constraints. The penalty models are exact if the penalty parameter is sufficiently large but finite. A practical penalty algorithm with postprocessing is then developed to approximately solve a series of subproblems with nonnegative and multiple spherical constraints. We study the asymptotic convergence and establish that any limit point is a weakly stationary point of the original problem and becomes a stationary point under some additional mild conditions. Extensive numerical results on the problem of computing the orthogonal projection onto nonnegative orthogonality constraints, the orthogonal nonnegative matrix factorization problems and the K-indicators model show the effectiveness of our proposed approach.